The sum of two numbers is $47$, and their difference is $19$. What are the two numbers?
Solution: Let $x$ be the first number, and let $y$ be the second number. The system of equations is: ${x+y = 47}$ ${x-y = 19}$ Solve for $x$ and $y$ using elimination. Add the top and bottom equations together. $ 2x = 66 $ $ x = \dfrac{66}{2} $ ${x = 33}$ Now that you know ${x = 33}$ , plug it back into $ {x+y = 47}$ to find $y$ ${(33)}{ + y = 47}$ ${y = 14}$ You can also plug ${x = 33}$ into $ {x-y = 19}$ and get the same answer for $y$ ${(33)}{ - y = 19}$ ${y = 14}$ Therefore, the larger number is $33$, and the smaller number is $14$.